Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

Abstract

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of $p$-Laplace type ($1 < p< \infty$) with strong absorption condition: $$ -\text{div}\,(\Phi(x, u, \nabla u)) + \lambda_0(x) u_{+}^q(x) = 0 \quad \text{in} \quad \Omega \subset \mathbb{R}^N, $$ where $\Phi: \Omega \times \mathbb{R}_{+} \times \mathbb{R}^N \to \mathbb{R}^N$ is a vector field with an appropriate $p$-structure, $\lambda_0$ is a non-negative and bounded function and $0\leq q<p-1$. Such a model is mathematically relevant because permits existence of solutions with dead core zones, i.e, \textit{a priori} unknown regions where non-negative solutions vanish identically. We establish sharp and improved $C^{\gamma}$ regularity estimates along free boundary points, namely $\mathfrak{F}_0(u, \Omega) = \partial \{u>0\} \cap \Omega$, where the regularity exponent is given explicitly by $\gamma = \frac{p}{p-1-q} \gg 1$. Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of $(N-1)$-Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the $p$-Laplace operator $-\Delta_p u + \lambda_0 u^q\chi_{\{u>0\}} = 0$ for any $\lambda_0>0$. \newline \newline \noindent \textbf{Keywords:} Quasi-linear elliptic operators of $p$-Laplace type, improved regularity estimates, Free boundary problems of dead core type, Liouville type results, Hausdorff measure estimates.